Questions C4 (1162 questions)

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Edexcel C4 Q1
  1. Use the substitution \(u = 4 + 3 x ^ { 2 }\) to find the exact value of
$$\int _ { 0 } ^ { 2 } \frac { 2 x } { \left( 4 + 3 x ^ { 2 } \right) ^ { 2 } } d x$$
\includegraphics[max width=\textwidth, alt={}]{a5902f63-b19f-4e37-94b8-35c3b47ab9de-03_2546_1815_88_158}
\begin{center} \begin{tabular}{|l|l|} \hline \multirow[b]{2}{*}{\begin{tabular}{l}
Edexcel C4 Q3
3. \(f ( x ) = \frac { 1 + 14 x } { ( 1 - x ) ( 1 + 2 x ) } , \quad | x | < \frac { 1 } { 2 } .\)
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
    (3)
  2. Hence find the exact value of \(\int _ { \frac { 1 } { 6 } } ^ { \frac { 1 } { 3 } } \mathrm { f } ( x ) \mathrm { d } x\), giving your answer in the form \(\ln p\), where \(p\) is rational.
    (5)
  3. Use the binomial theorem to expand \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying each term.
    (5)
    \end{tabular}} & Leave blank
    \hline &
    \hline \end{tabular} \end{center}
    1. continued
    2. The line \(l _ { 1 }\) has vector equation \(\mathbf { r } = \left( \begin{array} { c } 11
      5
      6 \end{array} \right) + \lambda \left( \begin{array} { l } 4
      2
      4 \end{array} \right)\), where \(\lambda\) is a parameter.
    The line \(l _ { 2 }\) has vector equation \(\mathbf { r } = \left( \begin{array} { c } 24
    4
    13 \end{array} \right) + \mu \left( \begin{array} { l } 7
    1
    5 \end{array} \right)\), where \(\mu\) is a parameter.
  4. Show that the lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
  5. Find the coordinates of their point of intersection. Given that \(\theta\) is the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\),
  6. find the value of \(\cos \theta\). Give your answer in the form \(k \sqrt { } 3\), where \(k\) is a simplified fraction.
Edexcel C4 Q5
5. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{a5902f63-b19f-4e37-94b8-35c3b47ab9de-08_497_919_270_635}
\end{figure} The curve shown in Fig. 1 has parametric equations $$x = \cos t , y = \sin 2 t , \quad 0 \leq t < 2 \pi .$$
  1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of the parameter \(t\).
  2. Find the values of the parameter \(t\) at the points where \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\).
  3. Hence give the exact values of the coordinates of the points on the curve where the tangents are parallel to the \(x\)-axis.
  4. Show that a cartesian equation for the part of the curve where \(0 \leq t < \pi\) is $$y = 2 x \sqrt { } \left( 1 - x ^ { 2 } \right)$$
  5. Write down a cartesian equation for the part of the curve where \(\pi \leq t < 2 \pi\).
    5. continued
Edexcel C4 Q6
6. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{a5902f63-b19f-4e37-94b8-35c3b47ab9de-10_579_1326_268_423}
\end{figure} Figure 2 shows the curve with equation $$y = x ^ { 2 } \sin \left( \frac { 1 } { 2 } x \right) , \quad 0 < x \leq 2 \pi .$$ The finite region \(R\) bounded by the line \(x = \pi\), the \(x\)-axis, and the curve is shown shaded in Fig 2.
  1. Find the exact value of the area of \(R\), by integration. Give your answer in terms of \(\pi\). The table shows corresponding values of \(x\) and \(y\).
    \(x\)\(\pi\)\(\frac { 5 \pi } { 4 }\)\(\frac { 3 \pi } { 2 }\)\(\frac { 7 \pi } { 4 }\)\(2 \pi\)
    \(y\)9.869614.24715.702\(G\)0
  2. Find the value of \(G\).
  3. Use the trapezium rule with values of \(x ^ { 2 } \sin \left( \frac { 1 } { 2 } x \right)\)
    1. at \(x = \pi , x = \frac { 3 \pi } { 2 }\) and \(x = 2 \pi\) to find an approximate value for the area \(R\), giving your answer to 4 significant figures,
    2. at \(x = \pi , x = \frac { 5 \pi } { 4 } , x = \frac { 3 \pi } { 2 } , x = \frac { 7 \pi } { 4 }\) and \(x = 2 \pi\) to find an improved approximation for the area \(R\), giving your answer to 4 significant figures.
      6. continued
Edexcel C4 Q7
7. In an experiment a scientist considered the loss of mass of a collection of picked leaves. The mass \(M\) grams of a single leaf was measured at times \(t\) days after the leaf was picked. The scientist attempted to find a relationship between \(M\) and \(t\). In a preliminary model she assumed that the rate of loss of mass was proportional to the mass \(M\) grams of the leaf.
  1. Write down a differential equation for the rate of change of mass of the leaf, using this model.
  2. Show, by differentiation, that \(M = 10 ( 0.98 ) ^ { t }\) satisfies this differential equation. Further studies implied that the mass \(M\) grams of a certain leaf satisfied a modified differential equation $$10 \frac { \mathrm {~d} M } { \mathrm {~d} t } = - k ( 10 M - 1 )$$ where \(k\) is a positive constant and \(t \geq 0\).
    Given that the mass of this leaf at time \(t = 0\) is 10 grams, and that its mass at time \(t = 10\) is 8.5 grams,
  3. solve the modified differential equation (I) to find the mass of this leaf at time \(t = 15\).
    7. continued
Edexcel C4 Specimen Q1
Use the binomial theorem to expand \(( 4 - 3 x ) ^ { - \frac { 1 } { 2 } }\), in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\). Give each coefficient as a simplified fraction.
Edexcel C4 Specimen Q3
3. Use the substitution \(x = \tan \theta\) to show that $$\int _ { 0 } ^ { 1 } \frac { 1 } { \left( 1 + x ^ { 2 } \right) ^ { 2 } } \mathrm {~d} x = \frac { \pi } { 8 } + \frac { 1 } { 4 }$$ (8)
Edexcel C4 Specimen Q4
4. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{0191bf56-a59e-44fe-af8c-bad796156f63-3_458_1552_415_223}
\end{figure} Figure 1 shows part of the curve with parametric equations $$x = \tan t , \quad y = \sin 2 t , \quad - \frac { \pi } { 2 } < t < \frac { \pi } { 2 } .$$
  1. Find the gradient of the curve at the point \(P\) where \(t = \frac { \pi } { 3 }\).
  2. Find an equation of the normal to the curve at \(P\).
  3. Find an equation of the normal to the curve at the point \(Q\) where \(t = \frac { \pi } { 4 }\).
Edexcel C4 Specimen Q5
5. The vector equations of two straight lines are $$\begin{aligned} & \mathbf { r } = 5 \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } ) \quad \text { and }
& \mathbf { r } = 2 \mathbf { i } - 11 \mathbf { j } + a \mathbf { k } + \mu ( - 3 \mathbf { i } - 4 \mathbf { j } + 5 \mathbf { k } ) . \end{aligned}$$ Given that the two lines intersect, find
  1. the coordinates of the point of intersection,
  2. the value of the constant \(a\),
  3. the acute angle between the two lines.
Edexcel C4 Specimen Q6
6. Given that $$\frac { 11 x - 1 } { ( 1 - x ) ^ { 2 } ( 2 + 3 x ) } \equiv \frac { A } { ( 1 - x ) ^ { 2 } } + \frac { B } { ( 1 - x ) } + \frac { C } { ( 2 + 3 x ) }$$
  1. find the values of \(A , B\) and \(C\).
  2. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 11 x - 1 } { ( 1 - x ) ^ { 2 } ( 2 + 3 x ) } \mathrm { d } x\), giving your answer in the form \(k + \ln a\), where \(k\) is an integer and \(a\) is a simplified fraction.
Edexcel C4 Specimen Q7
7. (a) Given that \(u = \frac { x } { 2 } - \frac { 1 } { 8 } \sin 4 x\), show that \(\frac { \mathrm { d } u } { \mathrm {~d} x } = \sin ^ { 2 } 2 x\). \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{0191bf56-a59e-44fe-af8c-bad796156f63-5_697_1239_587_367}
\end{figure} Figure 2 shows the finite region bounded by the curve \(y = x ^ { \frac { 1 } { 2 } } \sin 2 x\), the line \(x = \frac { \pi } { 4 }\) and the \(x\)-axis. This region is rotated through \(2 \pi\) radians about the \(x\)-axis.
(b) Using the result in part (a), or otherwise, find the exact value of the volume generated.
(8)
Edexcel C4 Specimen Q8
8. A circular stain grows in such a way that the rate of increase of its radius is inversely proportional to the square of the radius. Given that the area of the stain at time \(t\) seconds is \(A \mathrm {~cm} ^ { 2 }\),
  1. show that \(\frac { \mathrm { d } A } { \mathrm {~d} t } \propto \frac { 1 } { \sqrt { A } }\).
    (6) Another stain, which is growing more quickly, has area \(S \mathrm {~cm} ^ { 2 }\) at time \(t\) seconds. It is given that $$\frac { \mathrm { d } S } { \mathrm {~d} t } = \frac { 2 \mathrm { e } ^ { 2 t } } { \sqrt { S } }$$ Given that, for this second stain, \(S = 9\) at time \(t = 0\),
  2. solve the differential equation to find the time at which \(S = 16\). Give your answer to 2 significant figures. \section*{END}
Edexcel C4 2009 January Q2
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a5579938-e202-4543-8513-6483ede49850-03_410_552_205_694} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows part of the curve \(y = \frac { 3 } { \sqrt { } ( 1 + 4 x ) }\). The region \(R\) is bounded by the curve, the \(x\)-axis, and the lines \(x = 0\) and \(x = 2\), as shown shaded in Figure 1.
  1. Use integration to find the area of \(R\). The region \(R\) is rotated \(360 ^ { \circ }\) about the \(x\)-axis.
  2. Use integration to find the exact value of the volume of the solid formed.
Edexcel C4 2013 June Q2
  1. Use the binomial expansion to show that $$\left. \sqrt { ( } \frac { 1 + x } { 1 - x } \right) \approx 1 + x + \frac { 1 } { 2 } x ^ { 2 } , \quad | x | < 1$$
  2. Substitute \(x = \frac { 1 } { 26 }\) into $$\sqrt { \left( \frac { 1 + x } { 1 - x } \right) = 1 + x + \frac { 1 } { 2 } x ^ { 2 } }$$ to obtain an approximation to \(\sqrt { } 3\)
    Give your answer in the form \(\frac { a } { b }\) where \(a\) and \(b\) are integers.
Edexcel C4 Specimen Q2
The curve \(C\) has equation $$13 x ^ { 2 } + 13 y ^ { 2 } - 10 x y = 52$$ Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) as a function of \(x\) and \(y\), simplifying your answer.
(6)
OCR C4 2006 January Q1
1 Simplify \(\frac { x ^ { 3 } - 3 x ^ { 2 } } { x ^ { 2 } - 9 }\).
OCR C4 2006 January Q2
2 Given that \(\sin y = x y + x ^ { 2 }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
OCR C4 2006 January Q3
3
  1. Find the quotient and the remainder when \(3 x ^ { 3 } - 2 x ^ { 2 } + x + 7\) is divided by \(x ^ { 2 } - 2 x + 5\).
  2. Hence, or otherwise, determine the values of the constants \(a\) and \(b\) such that, when \(3 x ^ { 3 } - 2 x ^ { 2 } + a x + b\) is divided by \(x ^ { 2 } - 2 x + 5\), there is no remainder.
OCR C4 2006 January Q4
4
  1. Use integration by parts to find \(\int x \sec ^ { 2 } x \mathrm {~d} x\).
  2. Hence find \(\int x \tan ^ { 2 } x \mathrm {~d} x\).
OCR C4 2006 January Q5
5 A curve is given parametrically by the equations \(x = t ^ { 2 } , y = 2 t\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\), giving your answer in its simplest form.
  2. Show that the equation of the tangent to the curve at \(\left( p ^ { 2 } , 2 p \right)\) is $$p y = x + p ^ { 2 } .$$
  3. Find the coordinates of the point where the tangent at \(( 9,6 )\) meets the tangent at \(( 25 , - 10 )\).
OCR C4 2006 January Q6
6
  1. Show that the substitution \(x = \sin ^ { 2 } \theta\) transforms \(\int \sqrt { \frac { x } { 1 - x } } \mathrm {~d} x\) to \(\int 2 \sin ^ { 2 } \theta \mathrm {~d} \theta\).
  2. Hence find \(\int _ { 0 } ^ { 1 } \sqrt { \frac { x } { 1 - x } } \mathrm {~d} x\).
OCR C4 2006 January Q7
7 The expression \(\frac { 11 + 8 x } { ( 2 - x ) ( 1 + x ) ^ { 2 } }\) is denoted by \(\mathrm { f } ( x )\).
  1. Express \(\mathrm { f } ( x )\) in the form \(\frac { A } { 2 - x } + \frac { B } { 1 + x } + \frac { C } { ( 1 + x ) ^ { 2 } }\), where \(A , B\) and \(C\) are constants.
  2. Given that \(| x | < 1\), find the first 3 terms in the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\).
OCR C4 2006 January Q8
8
  1. Solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 - x } { y - 3 }$$ giving the particular solution that satisfies the condition \(y = 4\) when \(x = 5\).
  2. Show that this particular solution can be expressed in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$ where the values of the constants \(a , b\) and \(k\) are to be stated.
  3. Hence sketch the graph of the particular solution, indicating clearly its main features.
OCR C4 2006 January Q9
9 Two lines have vector equations $$\mathbf { r } = \left( \begin{array} { r } 4
2
- 6 \end{array} \right) + t \left( \begin{array} { r } - 8
1
- 2 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { r } - 2
a
- 2 \end{array} \right) + s \left( \begin{array} { r } - 9
2
- 5 \end{array} \right) ,$$ where \(a\) is a constant.
  1. Calculate the acute angle between the lines.
  2. Given that these two lines intersect, find \(a\) and the point of intersection.
OCR C4 2007 January Q1
1 It is given that $$f ( x ) = \frac { x ^ { 2 } + 2 x - 24 } { x ^ { 2 } - 4 x } \quad \text { for } x \neq 0 , x \neq 4$$ Express \(\mathrm { f } ( x )\) in its simplest form.